The process of getting digital photos printed works in exactly the same way as getting prints developed from 35mm film, except that instead of handing over a roll of film to the smiling employee behind the counter, you'll need to take in the camera's memory card.
Unlike a roll of exposed film, however, a memory card is reusable and is probably worth quite a bit of cash. So, rather than leaving your card at the shop overnight, most high-street mini-labs have a system of copying the digital image files from the card to the hard disk of the lab's computer so the card can be handed back to you straight away.